A368224 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k grid up to 180-degree rotation by an asymmetric tile.

1, 3, 3, 4, 10, 4, 10, 36, 36, 10, 16, 136, 256, 136, 16, 36, 528, 2080, 2080, 528, 36, 64, 2080, 16384, 32896, 16384, 2080, 64, 136, 8256, 131328, 524800, 524800, 131328, 8256, 136, 256, 32896, 1048576, 8390656, 16777216, 8390656, 1048576, 32896, 256

Offset

1,2

Links

Table of n, a(n) for n=1..45.

Peter Kagey, Illustration of T(3,3)=256

Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Example

Table begins:
  n\k|  1    2      3       4         5           6
  ---+---------------------------------------------
   1 |  1    3      4      10        16          36
   2 |  3   10     36     136       528        2080
   3 |  4   36    256    2080     16384      131328
   4 | 10  136   2080   32896    524800     8390656
   5 | 16  528  16384  524800  16777216   536887296
   6 | 36 2080 131328 8390656 536887296 34359869440

Mathematica

A368224[n_, m_] := 2^(n*m/2 - 1) (2^(n*m/2) + Boole[EvenQ[n*m]])

Crossrefs

Cf. A368220, A368222, A368223, A117401.

Sequence in context: A115284 A202869 A202871 * A144626 A033707 A178043

Adjacent sequences: A368221 A368222 A368223 * A368225 A368226 A368227

Keyword

nonn,tabl

Author

Peter Kagey, Dec 18 2023

Status

approved